Logarithmic potentials on $ {\mathbb{P}}^{n}$

Fatima Zahra Assila

doi:10.1016/j.crma.2018.02.004

Complex analysis

Logarithmic potentials on

P^{n}

[Potentiels logarithmiques sur

P^{n}

]

Présenté par : Jean-Pierre Demailly

Fatima Zahra Assila ¹

¹ Université Ibn Tofail, avenue de l'Université, Kénitra, Maroc

Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 283-287.

Résumé
Abstract

On étudie le potentiel logarithmique projectif $G_{μ}$ d'une mesure de probabilité μ sur l'espace projectif complexe $P^{n}$ . On établit que l'image de l'opérateur $μ ⟶ G_{μ}$ est contenue dans le domaine de définition (local) de l'opérateur de Monge–Ampère complexe agissant sur les fonctions quasi-plurisousharmoniques dans $P^{n}$ par rapport à la métrique de Fubini–Study. Si μ n'a pas d'atomes, on montre que la mesure de Monge–Ampère complexe du potentiel logarithmique de μ est absolument continue par rapport à la forme volume de Fubini–Study de $P^{n}$ .

We study the projective logarithmic potential $G_{μ}$ of a probability measure μ on the complex projective space $P^{n}$ . We prove that the range of the operator $μ ⟶ G_{μ}$ is contained in the (local) domain of definition of the complex Monge–Ampère operator acting on the class of quasi-plurisubharmonic functions on $P^{n}$ with respect to the Fubini–Study metric. Moreover, when the measure μ has no atom, we show that the complex Monge–Ampère measure of its logarithmic potential is an absolutely continuous measure with respect to the Fubini–Study volume form on $P^{n}$ .

Reçu le : 2017-06-23
Accepté le : 2018-02-12
Publié le : 2018-02-21

DOI : 10.1016/j.crma.2018.02.004

Affiliations des auteurs :

Fatima Zahra Assila ¹

¹ Université Ibn Tofail, avenue de l'Université, Kénitra, Maroc

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     author = {Fatima Zahra Assila},
     title = {Logarithmic potentials on $ {\mathbb{P}}^{n}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {283--287},
     publisher = {Elsevier},
     volume = {356},
     number = {3},
     year = {2018},
     doi = {10.1016/j.crma.2018.02.004},
     language = {en},
}

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Fatima Zahra Assila. Logarithmic potentials on $ {\mathbb{P}}^{n}$. Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 283-287. doi : 10.1016/j.crma.2018.02.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.02.004/

Bibliographie
Cité par

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